For {M_i}_(i element I) a family of R-modules indexed by a directed set I, let σ_(i j) :M_i->M_j i<=j be an R-module homomorphism. Call (M_i, σ_(i j)) a direct system over I when 1.σ_(i i)(m) = m for all m element M_i. 2. When i<=j<=k, σ_(i k) = σ_(j k) °σ_(i j).

directed set | inverse system | module | module homomorphism